As Bear and Ragged indicated, Groundspeak's Geocaching app should be able to download PQ data directly from the server, using the API. (FWIW, Neongeo and other third-party API-based can do the same thing.)

If you have the appropriate USB cable, then you can transfer GPX files from your PC to your phone. You'll need to confirm the connection on the phone, and then the phone will appear as a filesystem on your PC. From there, you can drag-and-drop the GPX file. Then you just need to "eject" the phone's filesystem from the PC, and disconnect the USB cable.

No. It's still 'in the shop' and can be downloaded again, without a fee.

Wonder if the "Run this query once then delete it" part is causing the problem?
Try running the PQ with "Uncheck the day of the week after the query runs" rather than "Delete"
(Vague memories of it being a problem in the past... Not for me, but did cause problems for others.)

When you start the app the start screen -with Groundspeak.com- has the version in the top right corner.
Only visible for a few seconds!

Quote

"Does it show in the second tab?"
Heading on second tab is: "Pocket queries ready for download (1)"

That second tab, should show the PQ if it's been run and generated.
IF the PQ is listed there, the phone app should see it.